Finite Group Databasebeta

← #29595518

Subgroups — #29595518

Order 15616.

1146 conjugacy classes of subgroups; only the 123 normal subgroups are shown.

normal subgroups only

This group has 1146 conjugacy classes of subgroups — too many to draw. Only its 123 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#14519766C211yesmaximal, characteristic
2C2×#16839558C214yesmaximal, complemented
3#21445743C211yesmaximal, characteristic, complemented
4C61×#1784C211yesmaximal, characteristic, complemented
5C23×D244C411yescharacteristic
6C2×#14519766—28no—
7C2×#14519766—21no—
8C2×#14519766—21no—
9C2×#14519766C412yes—
10C2×#14519766C2212yes—
11C22×#11758198—21no—
12C22×#11758198—21no—
13C22×C4×Dic61C411yescharacteristic
14C2×#14519696—21no—
15C2×#14519696C2212yescomplemented
16#16839726—21no—
17#16839558—28no—
18C23×Q8×C61C2211yescharacteristic
19C2×#101×C61—21no—
20C2×#101×C61C2212yescomplemented
21C61×#407—21no—
22C24×D61—21no—
23C22×D244—21no—
24C22×D244—24no—
25C22×D244—42no—
26C22×D244—42no—
27C22×D244C2×C412yes—
28C23×Dic61—21no—
29C2×#11758194—22no—
30#14519766—28no—
31#14519766—28no—
32#14519766—24no—
33#14519766—24no—
34#14519766—24no—
35#14519766D418yes—
36C2×#11758198—28no—
37C2×#11758198—28no—
38C2×#11758198—22no—
39C2×#11758198—22no—
40C2×#11758198—22no—
41C2×#11758198—22no—
42C2×C4×Dic61—21no—
43C2×C4×Dic61—24no—
44C2×C4×Dic61C2×C412yes—
45#14519696—24no—
46#14519696—42no—
47C22⋊#92418—42no—
48C22×Q8×C61—24no—
49C22×Q8×C61—42no—
50C22×Q8×C61—42no—
51C22×Q8×C61C2×C411yescharacteristic
52C22×Q8×C61D411yescharacteristic
53C22×Q8×C61D411yescharacteristic
54C22×Q8×C61C2311yescharacteristic
55C23×C4×C61C2×C411yescharacteristic
56C23×C4×C61D411yescharacteristic
57C23×C4×C61D411yescharacteristic
58C2×M16×C61—22no—
59#101×C61—24no—
60#101×C61—42no—
61C22⋊C8×C61—42no—
62C24×C61C22⋊C411yescharacteristic
63C23×D61—24no—
64C23×D61—22no—
65C23×D61—42no—
66C23×D61—48no—
67C23×D61—42no—
68C2×Q8×C61—22no—
69C2×Q8×C61—22no—
70C2×Q8×C61—22no—
71C2×Q8×C61—22no—
72C2×Q8×C61—22no—
73C2×Q8×C61—41no—
74C2×Q8×C61—41no—
75C2×Q8×C61—41no—
76C2×Q8×C61—41no—
77C2×Q8×C61—41no—
78C2×Q8×C61—41no—
79C2×Q8×C61—416no—
80C2×Q8×C61C2×D414yes—
81C22×C4×C61—22no—
82C22×C4×C61—22no—
83C22×C4×C61—22no—
84C22×C4×C61—22no—
85C22×C4×C61—22no—
86C22×C4×C61—22no—
87C22×C4×C61—22no—
88C22×C4×C61—41no—
89C22×C4×C61—41no—
90C22×C4×C61C22⋊C411yescharacteristic
91C22×C4×C61C22⋊C411yescharacteristic
92C22×C4×C61C22⋊C411yescharacteristic
93C22×C4×C61C22×C411yescharacteristic
94C22×C4×C61C2×D411yescharacteristic
95C22×C4×C61C2×D411yescharacteristic
96C22×Dic61—24no—
97C22×Dic61—22no—
98C22×Dic61—44no—
99C2×D244—24no—
100C2×D244—22no—
101C2×D244—416no—
102C2×D244—42no—
103C2×D244—42no—
104C2×D244—42no—
105C2×D244—42no—
106C2×D244C22⋊C414yes—
107M16×C61—42no—
108M16×C61—42no—
109C2×C8×C61—42no—
110#11758198—28no—
111#11758198—28no—
112#11758198—28no—
113#11758198—28no—
114C4×Dic61—24no—
115C4×Dic61—22no—
116C4×Dic61C22⋊C414yes—
117#11758194—42no—
118#11758194—42no—
119C2×#92418—42no—
120C23×C61—24no—
121C23×C61—21no—
122C23×C61—21no—
123C23×C61#9912yes—
124C23×C61C2×C22⋊C411yescharacteristic
125C22×D61—24no—
126C22×D61—48no—
127C22×D61—416no—
128C22×D61—48no—
129C22×D61—42no—
130C22×D61—42no—
131C22×D61—88no—
132C22×D61—84no—
133C22×D61—84no—
134Q8×C61—44no—
135Q8×C61—44no—
136Q8×C61—44no—
137Q8×C61—44no—
138C2×C4×C61—28no—
139C2×C4×C61—21no—
140C2×C4×C61—21no—
141C2×C4×C61—21no—
142C2×C4×C61—21no—
143C2×C4×C61—21no—
144C2×C4×C61—21no—
145C2×C4×C61—21no—
146C2×C4×C61—21no—
147C2×C4×C61—21no—
148C2×C4×C61—21no—
149C2×C4×C61—21no—
150C2×C4×C61—21no—
151C2×C4×C61—21no—
152C2×C4×C61—21no—
153C2×C4×C61—44no—
154C2×C4×C61—44no—
155C2×C4×C61#9912yes—
156C2×C4×C61C2×C22⋊C412yes—
157C2×C4×C61#12012yes—
158C2×C4×C61#12012yes—
159C2×Dic61—24no—
160C2×Dic61—48no—
161C2×Dic61—44no—
162D244—44no—
163D244—48no—
164D244—44no—
165C8×C61—82no—
166#92418—82no—
167#12699—611nomaximal, Sylow
168C22×C61—24no—
169C22×C61—24no—
170C22×C61—22no—
171C22×C61—22no—
172C22×C61—22no—
173C22×C61#35514yescomplemented
174C22×C61#37911yescharacteristic
175C22×C61C2×#9911yescharacteristic
176C22×C61C2×#9911yescharacteristic
177C2×D61—48no—
178C2×D61—44no—
179C2×D61—816no—
180C2×D61—84no—
181C2×D61—84no—
182C4×C61—22no—
183C4×C61—22no—
184C4×C61—22no—
185C4×C61—22no—
186C4×C61—41no—
187C4×C61—41no—
188Dic61—44no—
189C22×#124—611no—
190C2×#355—614no—
191#1784—611no—
192#1784D6111yescharacteristic, complemented
193C2×C61—21no—
194C2×C61—24no—
195C2×C61—21no—
196C2×C61#178111yescharacteristic
197C2×C61C2×#35512yes—
198D61—84no—
199C23×Q8C2×D6111yescharacteristic
200C23×D4—611no—
201C2×#124—612no—
202C2×#124—612no—
203C2×#124—1228no—
204C2×#124—1221no—
205C2×#124—1221no—
206C22×C22⋊C4—1221no—
207C22×C22⋊C4—1221no—
208C22×C42—611no—
209C2×#101—21no—
210C2×#101—612no—
211C2×#101—1221no—
212C2×#101C2×D6112yescomplemented
213#407—21no—
214#407—1221no—
215#355—1228no—
216C61#1269911yescharacteristic, complemented, Sylow
217C25—1221no—
218C22×Q8—24no—
219C22×Q8—42no—
220C22×Q8—42no—
221C22×Q8C4×D6111yescharacteristic
222C22×Q8D24411yescharacteristic
223C22×Q8C22⋊D6111yescharacteristic
224C22×Q8C22×D6111yescharacteristic
225C22×D4—612no—
226C22×D4—1224no—
227C22×D4—1221no—
228C22×D4—2442no—
229C22×D4—2442no—
230C23×C4—1221no—
231C23×C4C4×D6111yescharacteristic
232C23×C4D24411yescharacteristic
233C23×C4C22⋊D6111yescharacteristic
234C2×M16—22no—
235C2×M16—1222no—
236#124—618no—
237#124—1224no—
238#124—1224no—
239#124—1228no—
240#124—1228no—
241#124—1224no—
242C2×C22⋊C4—1228no—
243C2×C22⋊C4—1228no—
244C2×C22⋊C4—1222no—
245C2×C22⋊C4—1222no—
246C2×C22⋊C4—1222no—
247C2×C22⋊C4—1222no—
248C2×C42—612no—
249C2×C42—1224no—
250C2×C42—1221no—
251#101—24no—
252#101—42no—
253#101—1224no—
254#101—2442no—
255C22⋊C8—42no—
256C22⋊C8—2442no—
257C24—1224no—
258C24—1222no—
259C24—2442no—
260C24—2448no—
261C24—2442no—
262C24#1175819811yescharacteristic
263C2×Q8—22no—
264C2×Q8—22no—
265C2×Q8—22no—
266C2×Q8—22no—
267C2×Q8—22no—
268C2×Q8—41no—
269C2×Q8—41no—
270C2×Q8—41no—
271C2×Q8—41no—
272C2×Q8—41no—
273C2×Q8—41no—
274C2×Q8—416no—
275C2×Q8D4×D6114yes—
276C2×D4—614no—
277C2×D4—1224no—
278C2×D4—1222no—
279C2×D4—2442no—
280C2×D4—2442no—
281C2×D4—24416no—
282C2×D4—2442no—
283C2×D4—2442no—
284C22×C4—22no—
285C22×C4—22no—
286C22×C4—22no—
287C22×C4—22no—
288C22×C4—22no—
289C22×C4—22no—
290C22×C4—22no—
291C22×C4—41no—
292C22×C4—41no—
293C22×C4—1224no—
294C22×C4—1222no—
295C22×C4—2444no—
296C22×C4#1175819811yescharacteristic
297C22×C4#1175819811yescharacteristic
298C22×C4#1175819811yescharacteristic
299C22×C4C2×C4×D6111yescharacteristic
300C22×C4C2×D24411yescharacteristic
301C22×C4C2×C22⋊D6111yescharacteristic
302M16—42no—
303M16—42no—
304M16—2442no—
305M16—2442no—
306C2×C8—42no—
307C2×C8—2442no—
308C22⋊C4—1228no—
309C22⋊C4—1228no—
310C22⋊C4—1228no—
311C22⋊C4—1228no—
312C42—614no—
313C42—1224no—
314C42—1222no—
315C23—21no—
316C23—24no—
317C23—21no—
318C23—1224no—
319C23—4888no—
320C23—4884no—
321C23—4884no—
322C23—2448no—
323C23—24416no—
324C23—2448no—
325C23—2442no—
326C23—2442no—
327C23#1451969112yes—
328C23C2×#1175819811yescharacteristic
329Q8—44no—
330Q8—44no—
331Q8—44no—
332Q8—44no—
333D4—2444no—
334D4—2448no—
335D4—2444no—
336C2×C4—28no—
337C2×C4—21no—
338C2×C4—21no—
339C2×C4—21no—
340C2×C4—21no—
341C2×C4—21no—
342C2×C4—21no—
343C2×C4—21no—
344C2×C4—21no—
345C2×C4—21no—
346C2×C4—21no—
347C2×C4—21no—
348C2×C4—21no—
349C2×C4—21no—
350C2×C4—21no—
351C2×C4—44no—
352C2×C4—44no—
353C2×C4—1224no—
354C2×C4—2448no—
355C2×C4—2444no—
356C2×C4#1451969112yes—
357C2×C4C22⋊C4×D6112yes—
358C2×C4#1451974512yes—
359C2×C4#1451974712yes—
360C8—82no—
361C8—4882no—
362C22—24no—
363C22—24no—
364C22—22no—
365C22—22no—
366C22—22no—
367C22—4884no—
368C22—48816no—
369C22—4884no—
370C22—2448no—
371C22—2444no—
372C22#1683955814yescomplemented
373C22#1683961711yescharacteristic
374C22C2×#1451969111yescharacteristic
375C22C2×#1451969111yescharacteristic
376C4—22no—
377C4—22no—
378C4—22no—
379C4—22no—
380C4—41no—
381C4—41no—
382C4—2444no—
383C2—21no—
384C2—21no—
385C2—24no—
386C2—4884no—
387C2#2144573311yescharacteristic
388C2C2×#1683955812yes—
subgroupG/H
centerC22C2×#14519691
commutatorC22×C4×C61C22×C4
FrattiniC22×Q8C22×D61
FittingC61×#1784C2
socleC22×C61C2×#99
prime radicalC23×D244C4
cosocleC22×Q8×C61C23
hypercenter#1784D61
hypocenterC61#12699
LCSG ▷ C22×C4×C61 ▷ C22×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C22 ◁ C24 ◁ C23×Q8 ◁ #1784

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

NH ≅ G/Ntype
2. C2×#16839558C2semidirect
3. #21445743C2semidirect
4. C61×#1784C2semidirect
15. C2×#14519696C22semidirect
20. C2×#101×C61C22semidirect
173. C22×C61#355semidirect
192. #1784D61semidirect
212. C2×#101C2×D61semidirect
216. C61#12699semidirect
372. C22#16839558semidirect

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroupindex
C22×#145197662
C2×#168395582
#214457432
C61×#17842
#1269961